75.27.4 problem 779

Internal problem ID [17240]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 20. The method of elimination. Exercises page 212
Problem number : 779
Date solved : Tuesday, January 28, 2025 at 09:58:21 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+5 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )-3 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = -2\\ y \left (0\right ) = 1 \end{align*}

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 26

dsolve([diff(x(t),t) = x(t)+5*y(t), diff(y(t),t) = -x(t)-3*y(t), x(0) = -2, y(0) = 1], singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \left (-2 \cos \left (t \right )+\sin \left (t \right )\right ) \\ y \left (t \right ) &= \cos \left (t \right ) {\mathrm e}^{-t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 20

DSolve[{D[x[t],t]+3*x[t]+4*y[t]==0,D[y[t],t]+2*x[t]+5*y[t]==0},{x[0]==-2,y[0]==1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to -2 e^{-t} \\ y(t)\to e^{-t} \\ \end{align*}